Subgroups — #34005427

Order 18688.

378 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625864	C₂	1	1	yes	maximal, characteristic
2	#22625848	C₂	1	1	yes	maximal, characteristic
3	#22624761	C₂	1	2	yes	maximal
4	#22624761	C₂	1	2	yes	maximal
5	#97×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556658	—	2	2	no	—
7	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
8	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
9	#17989453	—	2	2	no	—
10	C₂×#14556502	C₄	1	1	yes	characteristic
11	C₂×#14556501	C₄	1	1	yes	characteristic
12	C₂×#14556501	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×#97×C₇₃	C₄	1	1	yes	characteristic
17	C₂²×#10596833	—	2	2	no	—
18	C₂²×#10596833	—	2	2	no	—
19	#14556658	—	4	4	no	—
20	C₄⋊#10596833	—	2	2	no	—
21	C₄⋊#10596833	—	2	2	no	—
22	C₄⋊#10596833	—	2	2	no	—
23	C₄⋊#10596833	—	2	2	no	—
24	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
25	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
28	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
29	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
30	#14556502	—	2	2	no	—
31	#14556501	—	2	2	no	—
32	#14556501	C₈	1	4	yes	complemented
33	C₈×Dic₇₃	—	2	1	no	—
34	C₈×Dic₇₃	—	2	1	no	—
35	C₈×Dic₇₃	—	2	1	no	—
36	C₈×Dic₇₃	—	2	1	no	—
37	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
40	#97×C₇₃	—	2	2	no	—
41	C₂×#10596833	—	4	2	no	—
42	C₂×#10596833	—	4	2	no	—
43	C₂×#10596833	—	4	2	no	—
44	C₂×#10596833	—	4	4	no	—
45	C₂×#10596833	—	4	2	no	—
46	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
47	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×#10596831	—	2	1	no	—
50	C₂×#10596831	—	2	1	no	—
51	C₂×#10596831	—	2	1	no	—
52	C₂×#10596831	—	2	1	no	—
53	C₂×#10596831	C₂×C₈	1	2	yes	—
54	C₂×#10596831	M₁₆	1	2	yes	—
55	C₄×Dic₇₃	—	2	2	no	—
56	C₄×Dic₇₃	—	2	2	no	—
57	C₄×Dic₇₃	—	4	2	no	—
58	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
59	C₄×Dic₇₃	M₁₆	1	2	yes	—
60	C₄×Dic₇₃	#41	1	1	yes	characteristic
61	C₄×Dic₇₃	#41	1	1	yes	characteristic
62	C₄×Dic₇₃	#41	1	1	yes	characteristic
63	C₄×Dic₇₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₃	—	2	1	no	—
67	C₂×C₈×C₇₃	—	2	1	no	—
68	C₂×C₈×C₇₃	—	2	1	no	—
69	C₂×C₈×C₇₃	—	2	1	no	—
70	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
71	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
72	C₄²×C₇₃	—	2	1	no	—
73	C₄²×C₇₃	—	2	1	no	—
74	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	—	2	2	no	—
76	C₂×C₄×C₇₃	—	2	2	no	—
77	C₂×C₄×C₇₃	—	2	2	no	—
78	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
79	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×Dic₇₃	—	2	2	no	—
85	C₂×Dic₇₃	—	2	2	no	—
86	C₂×Dic₇₃	—	2	2	no	—
87	C₂×Dic₇₃	C₄⋊C₈	1	4	yes	—
88	C₂×Dic₇₃	#124	1	1	yes	characteristic
89	C₂×Dic₇₃	#124	1	1	yes	characteristic
90	C₂×Dic₇₃	#125	1	1	yes	characteristic
91	C₂×Dic₇₃	#125	1	1	yes	characteristic
92	C₂×Dic₇₃	AGL(1,ℤ/8)	1	2	yes	—
93	C₂×Dic₇₃	#137	1	2	yes	—
94	#10596833	—	8	2	no	—
95	#10596833	—	8	2	no	—
96	C₈×C₇₃	—	2	2	no	—
97	C₈×C₇₃	—	2	2	no	—
98	#10596831	—	2	2	no	—
99	#10596831	—	2	2	no	—
100	C₂²×C₇₃	#382	1	1	yes	characteristic
101	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₂²×C₇₃	#423	1	1	yes	characteristic
103	C₂²×C₇₃	#428	1	2	yes	—
104	C₂²×C₇₃	#430	1	2	yes	—
105	C₄×C₇₃	—	4	2	no	—
106	C₄×C₇₃	#431	1	1	yes	characteristic
107	C₄×C₇₃	#431	1	1	yes	characteristic
108	C₄×C₇₃	#432	1	1	yes	characteristic
109	C₄×C₇₃	#432	1	1	yes	characteristic
110	Dic₇₃	—	2	4	no	—
111	Dic₇₃	—	2	1	no	—
112	Dic₇₃	—	2	1	no	—
113	Dic₇₃	#489	1	2	yes	—
114	Dic₇₃	#490	1	1	yes	characteristic
115	Dic₇₃	#491	1	1	yes	characteristic
116	#8786	—	73	1	no	maximal, Sylow
117	C₂×C₇₃	#1565	1	2	yes	—
118	C₂×C₇₃	#1569	1	2	yes	—
119	C₂×C₇₃	#1831	1	1	yes	characteristic
120	C₂×C₇₃	#1836	1	1	yes	characteristic
121	C₂×C₇₃	#1838	1	1	yes	characteristic
122	#1842	—	73	1	no	—
123	#1831	—	73	1	no	—
124	C₄×#97	—	73	1	no	—
125	#1277	—	73	2	no	—
126	#1277	—	73	2	no	—
127	C₇₃	#8786	1	1	yes	characteristic, complemented, Sylow
128	C₂×C₄⋊C₈	—	73	2	no	—
129	C₂×C₄⋊C₈	—	73	2	no	—
130	C₂×C₄⋊C₈	—	146	2	no	—
131	C₂×#97	—	73	1	no	—
132	C₂×#97	—	73	1	no	—
133	C₂×#97	—	73	1	no	—
134	C₂×#97	#64408	1	1	yes	characteristic
135	C₂×C₄×C₈	—	73	1	no	—
136	C₂×C₄×C₈	—	73	1	no	—
137	C₄³	—	73	1	no	—
138	#336	—	146	2	no	—
139	C₂²×C₈	—	73	1	no	—
140	C₂²×C₈	—	73	1	no	—
141	C₂²×C₈	—	146	2	no	—
142	C₂²×C₈	—	146	2	no	—
143	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
144	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
145	C₂×C₄²	—	73	2	no	—
146	C₂×C₄²	—	73	1	no	—
147	C₂×C₄²	—	73	1	no	—
148	C₂×C₄²	—	73	2	no	—
149	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
150	C₄⋊C₈	—	292	4	no	—
151	C₄⋊C₈	—	146	2	no	—
152	C₄⋊C₈	—	146	2	no	—
153	C₄⋊C₈	—	146	2	no	—
154	C₄⋊C₈	—	146	2	no	—
155	#97	—	2	2	no	—
156	#97	—	73	4	no	—
157	#97	—	146	2	no	—
158	#97	—	146	2	no	—
159	C₄×C₈	—	146	1	no	—
160	C₄×C₈	—	146	1	no	—
161	C₄×C₈	—	146	1	no	—
162	C₄×C₈	—	146	1	no	—
163	C₂²×C₄	—	73	1	no	—
164	C₂²×C₄	—	73	2	no	—
165	C₂²×C₄	—	73	1	no	—
166	C₂²×C₄	C₄⋊#64408	1	2	yes	—
167	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	292	4	no	—
173	C₂×C₈	—	292	2	no	—
174	C₂×C₈	—	292	2	no	—
175	C₂×C₈	—	292	2	no	—
176	C₂×C₈	—	292	2	no	—
177	C₂×C₈	—	73	2	no	—
178	C₂×C₈	—	73	2	no	—
179	C₂×C₈	—	146	1	no	—
180	C₂×C₈	—	146	1	no	—
181	C₂×C₈	—	146	1	no	—
182	C₂×C₈	—	146	1	no	—
183	C₂×C₈	#11924036	1	2	yes	—
184	C₂×C₈	#11924037	1	2	yes	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	292	2	no	—
188	C₄²	—	73	2	no	—
189	C₄²	—	73	2	no	—
190	C₄²	—	73	1	no	—
191	C₄²	—	73	1	no	—
192	C₄²	—	73	1	no	—
193	C₄²	—	73	1	no	—
194	C₄²	—	146	2	no	—
195	C₄²	—	146	2	no	—
196	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	73	1	no	—
201	C₂×C₄	—	73	1	no	—
202	C₂×C₄	—	73	2	no	—
203	C₂×C₄	—	73	2	no	—
204	C₂×C₄	—	73	1	no	—
205	C₂×C₄	—	73	1	no	—
206	C₂×C₄	—	73	4	no	—
207	C₂×C₄	—	146	2	no	—
208	C₂×C₄	—	146	2	no	—
209	C₂×C₄	—	146	2	no	—
210	C₂×C₄	#14556670	1	1	yes	characteristic
211	C₂×C₄	#14556670	1	1	yes	characteristic
212	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
213	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
214	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
215	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	2	2	no	—
218	C₈	—	584	2	no	—
219	C₈	—	584	2	no	—
220	C₈	—	146	2	no	—
221	C₈	—	146	2	no	—
222	C₂²	#17989531	1	1	yes	characteristic
223	C₂²	#17989588	1	1	yes	characteristic
224	C₂²	#17989590	1	1	yes	characteristic
225	C₂²	#17989598	1	2	yes	—
226	C₂²	#17989602	1	2	yes	—
227	C₄	—	4	2	no	—
228	C₄	—	73	1	no	—
229	C₄	—	73	2	no	—
230	C₄	—	73	1	no	—
231	C₄	—	146	1	no	—
232	C₄	—	146	1	no	—
233	C₄	—	146	4	no	—
234	C₄	#17989604	1	1	yes	characteristic
235	C₄	#17989605	1	1	yes	characteristic
236	C₄	#17989607	1	1	yes	characteristic
237	C₄	#17989610	1	1	yes	characteristic
238	C₂	#22625483	1	2	yes	—
239	C₂	#22625495	1	2	yes	—
240	C₂	#22625847	1	1	yes	characteristic
241	C₂	#22625854	1	1	yes	characteristic
242	C₂	#22625857	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#97×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#97	#64408
hypocenter	C₇₃	#8786
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
32. #14556501	C₈	semidirect
127. C₇₃	#8786	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22625864	2
#22625848	2
#22624761	2
#97×Dic₇₃	2
#8786	73